A Biological Basis For Gender Differences In Math?
    By News Staff | March 18th 2013 02:37 PM | 26 comments | Print | E-mail | Track Comments
    American education reform in the first decade of the 21st century, called No Child Left Behind, resulted in math parity among boys and girls for the first time in history.

    It may have taken so long because legacy education methods fight biology and not the persistent claims that there is subtle bias (hidden, stereotype threat and other sociological rationalizations), even in countries with high gender equality.  Sex differences in math and reading scores persisted in the 75 nations examined by a University of Missouri and University of Leeds study; girls consistently scored higher in reading, while boys got higher scores in math. 

    Instead of being gender-based, gaps are linked and vary with overall social and economic conditions of the nation. 

    “Educational systems could be improved by acknowledging that, in general, boys and girls are different,” said University of Missouri biologist David Geary in their statement. “For example, in trying to close the sex gap in math scores, the reading gap was left behind. Now, our study has found that the difference between girls’ and boys’ reading scores was three times larger than the sex difference in math scores. Girls’ higher scores in reading could lead to advantages in admissions to certain university programs, such as marketing, journalism or literature, and subsequently careers in those fields. Boys lower reading scores could correlate to problems in any career, since reading is essential in most jobs.”

    Generally, when conditions are good, the math gap increases and the reading gap decreases and when conditions are bad the math gap decreases and the reading gap increases. This pattern remained consistent within nations as well as among them, according to the work by Geary and Gijsbert Stoet of the University of Leeds that included testing performance data from 1.5 million 15-year-olds in 75 nations. 

    “In adult life, there are more male CEOs, but also more homeless men,” said Geary. “Boys’ prospects in life seem to react more intensely to positive and negative social conditions, hence we see more variation in boys’ testing scores, especially when conditions are bad.”

    The top five percent of scores within nations generally showed girls to be lower in math and boys to be lower in reading. That pattern continued in lower scoring groups until reaching the lowest scoring students, where the math achievement of boys and girls evened out but the reading gap increased, according to Geary.

    “The consistent pattern within nations suggests the sex differences are not simply related to socio-economic factors,” said Geary. Socio-economic and cultural factors are important in that they influence the performance of all students, but boys, as a group, respond more strongly than girls, perhaps due to a biological difference in sensitivity to wider conditions.” For example, in nations with impoverished or violent conditions, boys’ scores tended to fall faster and further than girls. On the other hand, in wealthier, socially stable nations boys’ scores benefitted more than girls. This resulted in boys reducing the reading gap and widening the math gap.

    Negative correlation between boys’ disadvantage in reading achievement (y-axis) and girls’ disadvantage in mathematics achievement. Each data point indicates the sex differences of one country. Positive values indicate a larger disadvantage, negative values an advantage. Red points indicate nations in which girls’ mathematics achievement is significantly higher than that of boys; blue points indicate nations in which boys’ mathematics achievement is significantly higher than that of girls; and, black points indicate nations in which there is no statistically significant difference in mathematics achievement. The advantage of girls in reading achievement is statistically significant in all nations, except for 2 in 2000 (Israel, Peru) and one in 2003 (Liechtenstein). Credit: doi:10.1371/journal.pone.0057988.g002

    “This finding has important implications for how we interpret the math gap of other countries,” said co-author Gijsbert Stoet of the University of Leeds. “For example, policy makers often take Sweden as an example of being particularly good for reducing the gender gap in science, technology, engineering and math, but they do not realize that Swedish boys fall behind in reading more so than in most other highly developed nations. This is a good example of the inverse relation between the math and reading gaps. This phenomenon urgently needs more attention.”

    Citation: Stoet G, Geary DC (2013) Sex Differences in Mathematics and Reading Achievement Are Inversely Related: Within- and Across-Nation Assessment of 10 Years of PISA Data. PLoS ONE 8(3): e57988. doi:10.1371/journal.pone.0057988


    So social studies once again confirm what people with perspicacity and common sense have witnessed every day for a few thousand years. Men and women are genetically NOT equivalent and cannot be made so through education.

    I believe the whole system is well designed to teach girls and is horrible at teaching boys.  Where the boys used to be able to more naturally learn the material (like math), they did better.  Make the system optimal to teach girls and the boys are no longer being educated.  Since most teachers are women they teach in the way that suits them, hence girls get an education more suited to them than the boys.   

    Does this surprise anyone?  Soon all the boys will be Lost Boys.  Maybe then the scientific community will accept the natural differences that exist between boys and girls, but I wouldn't bet on it.

    You are probably right about the fact that male and female brains differ not being accepted anytime soon.  The reasons are all politics.  In the memorable past science was used to justify systems of inequality against women, and socially disadvantaged groups.  So anything science says has to be said very carefully.
    Until we accept that male and female brains are different, as well as the spectrum of variances that can exist between them we will live in a psycho-social dark age.  One where men outnumber women in some fields by an absurd proportion, and women outnumber men in an absurd proportion.  Not because men and women are better at different task, but because we don't appreciate or idealize what women bring to so-called "men's task", and what men bring to so-called "women's task".    (A prime example would be childrearing.  In most of the USA, at least for a long time, it was assumed by all that divorced dads had no role other than to provide money, married dad's had not much more to do even if they wanted to.  We see now what a lack of fathers does to children.)
    Science advances as much by mistakes as by plans.
    The FAA did a study several years ago that showed that a mixed cockpit was potentially beneficial. Females tended to be much more thorough and meticulous in the preflight planning (language) and execution than their male counterparts. However male pilots tended to be better at dealing with unexpected emergency situations, especially when ad-hoc non-standard procedures, where quick spacial and temporal abstractions, are required. Think of US Air's Captain Scully choosing to land his hobbled jet into the Hudson River.

    These differing aspects tended to become washed out in the statistics, since the actual accident rates were very similar, but for very different reasons. The assertions were that the mixed team could compliment each other and thus cover each other's inherent weaknesses.

    The assertion was not "politically correct." Can you guess how long it took for well known idealistic and politically active special interest groups to successfully attack and suppress this research?

    In another research study males and females were intentionally disorientated and when let out of a dark container males will nearly always look up to the sky to get a general bearing on the Sun for direction in order to orientate themselves. While females will generally look around and initially make references to very local items such as trees, rocks, or whatever items happen to be available. One type of behavior is associated with hunting skills and the other is associated with gathering skills. Can you tell the which is which?

    Drop the "political correctness" and the answers become obvious. Political correctness is just one tool used very successfully by idealists in order to suppress the truth for the purposes of pushing forward their own personal agendas.

    Yet, even when one boils things down to hunting vs gathering.  The "male" skill you described could be beneficial in gathering.  The male might be able to get to the distant place where he once gathered some yummy berries.  The "female" skill you described could be helpful in hunting for the plentiful small game which are always around.   Then of course everyone is good at one and the other to a certain extent.  
    As for political correctness.  I find it is more often about supporting the agendas of those with a certain point of view.  One in which equality means sameness, in which diversity is a commonly said word, yet noticing the differences that make people diverse (even in the most respectful ways possible) is frowned on.  
    Science advances as much by mistakes as by plans.
    Gerhard Adam
    Most of what are described as "skills" are taught and have no biological basis in anything.  As for trying to tie psychological behaviors to such "skills" ... tough argument if the skills don't actually exist as an intrinsic trait.

    While there are certainly psychological differences between males and females, the hunter-gatherer argument is largely a "just-so" story and has little supporting evidence to make any conjectures.
    Mundus vult decipi
    "... tough argument if the skills don't actually exist as an intrinsic trait."

    So you must be a blank slater and think every skill demonstrated by any living creature is taught.

    The funny thing is that in almost every instance, where men have traditionally excelled, in order to "equalize the playing field" the standards for women must be lowered below those set for their male counterparts, i.e. firefighters.

    If they are not able to apply different standards then few women will ever rise to the skill levels necessary to even qualify to reach the minimum standards, for example sports. Physicalness aside I still root for Danika Patrick, if for no other reason that she is helping to break down the southern "good 'ol boy" system. I don't, however, ever expect her to actually win a high level race without a lot of help propping her up.

    Another example is Marion Rehume, a woman who made it onto a professional NHL mens hockey team as a goal tender. A huge accomplishment but, unfortunately, her career was rather short since in long term her taught "skills" were not compatible or up to the standards that would have allowed her to excel in this field along side of her male counterparts.

    The goal tending and race car driver positions are great examples since they do not normally have to compete one-to-one with the other players on a purely physical level.

    In yet another example a woman, I don't recall her name, made the cut for some professional mens golf tournaments but no women have yet won one of these competitions.

    These were very motivated and highly sponsored women who were "taught" by some of the best teachers in their fields, yet still don't rise to the level necessary for them to come close to dominate their fields. I must also add that these very exceptional women are also statistical outliers in their gender.

    Well here's the thing.  You say that accommodating women means lowering standards.  I'd say that it means recognizing the strength's that women can bring to a task.
    Suppose we are talking about fire fighting.  A woman should not be put in a position where they would be expected to carry a big heavy unconscious man by themselves.   Unless, of course, they are physically able to pass the same test as any other firefighter.  Likewise for men.  This does not make a woman a second class fire fighter.  There are going to be situations where a, most likely, smaller body is advantageous.  Such as having to get into tighter spaces.  Perhaps, due to a generally shorter height, being able to last longer in a fire.  After all, heat rises. In a fire the temp can vary by tens of degrees within a foot of height. 

    When it comes to child rearing, something thought of as inherently for women.  Beyond a certain biological need for milk, males bring certain skills to raising babies that females do not.  For example, a father baby sitting a toddler may be better able to keep up with their fast little legs than the mother.  Certain kinds of play are more likely to be done by a father than a mother,  (Mostly for reasons of gender identity training which are as Gearhard notes, taught). 

    The bottom line is that difference between men and women does not make one gender superior or the other inferior.  The fact that this is largely a matter of sexism is illustrated well by a story told by Ben Barres.  Ben is a Female to Male transman who works as a scientist.  He recalls being told by a colleague that was familiar with some of his work as a woman that "Your work is much better than your sisters."  Same person, same brain, same work, viewed as more competent due to presenting a masculine visage.  
    Science advances as much by mistakes as by plans.

    I couldn't agree with you more.

    I never implied that one side of the inequality of the sexes means in any way superiority. Your thoughts on child rearing are completely in line with the assertions of the FAA study. The differences between the sexes can, when complimentary, result in superiority. However, when the differences in the sexes are adversarial it will undoubtably result in an inferior situation.

    To deny that there are different areas where the differing sexes naturally excel is the crux of the philosophic fallacy of "gender equality."

    Is anyone else having problems with this web page throwing "service unavailable" errors?

    Gerhard Adam
    One doesn't have to be a blank slate.  All one has to do is observe whether those skills ever arise without teaching.  Any skill that is not intrinsic, cannot lay claim to genetics.  Therefore it is either taught, or acquired through experience.  Genetics may provide a framework in which it can be cultivated, but the skill doesn't exist without the teaching. 

    In matters of size and strength, then it should be clear that men will tend to be favored, when the contest is biased towards their strengths.  Again, this tells us little or nothing about actual human biology.  Let's note that strength is not a skill.  It is simply the genetic luck of the draw.  It's like claiming that being tall in basketball is a "talent".

    There's no question that there are gender differences, but your simple comparisons don't rise to the standard necessary to demonstrate that there's any kind of intrinsic superiority in one over the other.
    If they are not able to apply different standards then few women will ever rise to the skill levels necessary to even qualify to reach the minimum standards, for example sports...
    What evidence is there that they lack skill versus that they lack strength?

    Even Danika Patrick isn't ranked last, so what does that say about the males ranked below her?  Are we to simply conclude that this one woman must beat everyone's rankings before her position can be taken seriously?  That truly is a different standard, since it isn't one that you would apply to the majority of drivers.  
    Mundus vult decipi
    The differences between male and female brains have long been known to be very much steeped in biology. There are more connections between the two halves of the brain in females. It has been asserted that this facilitates the superior linguistic skills of women in general. It is also thought to be responsible for mens intrinsic superiority when the quick response is required to counter the random and often irrational activities of the actions of prey when pursued by the hunter.

    from a well respected publication:
    "Men’s greater systemizing and mechanistic skills are the primary reasons why they are better than women at mathematics, physics, and engineering, because all of these fields deal with various rational “systems” governed by rules. Women’s greater empathizing and mentalistic skills are the primary reasons why they are better at languages and why they are better judges of character. Women also dominate primatology, which, like mothering of infants, requires understanding and reading the minds of individuals with whom they cannot communicate by language."

    And from another well respected publication:
    "The early appearance of any sex difference suggests it is innately programmed—selected for through evolution and fixed into our behavioral development through either prenatal hormone exposure or early gene expression differences."

    I understand that, historically, you have always taken the adversarial position against nearly all of my comments in this blog site, I'm not at all surprised here either, I don't know your motives, nor do I really care.

    Gerhard Adam
    I understand that, historically, you have always taken the adversarial position against nearly all of my comments in this blog site, I'm not at all surprised here either, I don't know your motives, nor do I really care.
    Then why bring it up?  I don't single people out simply to oppose their comments.  I oppose their ideas.

    Your quote about sex differences isn't relevant, since the topic isn't about the psychological differences of being male or female, but rather the pertinent skills that they possess [specifically mathematical].

    The point is that you simply presume that the statements you make are true.  We already know they are not, because everything a man can do, invariably one can find a woman to do it also.  While you might be able to make a general argument if we are talking about averages, but then the average "skill" or standard wouldn't be met by those of the same gender either.  As soon as you introduce any kind of elite skill, the overwhelming majority of humans won't possess it [male or female].  So, it renders the biological argument moot, because it is clear that it is the elite athlete that is the statistical outlier, and not an indication of the biological norm.

    Certainly attitudes may be different between the genders, interests, cultural influences, etc. etc. etc.  However that doesn't translate into skills [i.e. the ability or inability to perform an action based on biology].

    Again, from the Psychology Today quote ... I don't want to hear pronouncements, I want to see data and it shouldn't come from examining less than 1% of the population.

    Once you eliminate size or strength [which would also eliminate most men], there isn't any skill intrinsic or otherwise that either gender can't acquire if they are so motivated and unconstrained.
    Mundus vult decipi
    The women are statistical outliers on the right side of the curve when viewed from their own gender perspective. However, when viewed inside of the outliers of all of the professional sports athletes, in the examples that I mentioned, the women are still outliers, this time on the left side of the curve.

    This is VERY significant.

    You are simply selecting what you choose to believe is true and selectively rejecting what doesn't fit your own world perspective. I don't have a problem with that, but it isn't at all scientific, in fact its a bit idealistic.

    Gerhard Adam
    I thought I was clear on eliminating strength and size?

    However, you are the one misunderstanding the key point.  These women are just as skillful as the men.  There is no situation of where a woman can't play basketball or baseball or hockey based on some biological block.  The "skill" exists and is readily usable.

    The only difference is in the size/strength elements and in the number of women that would choose to pursue such a career choice.  It should also be clear that with significantly fewer women pursuing such an objective [especially if there may not even be many opportunities], then the few that do manage it are held to a standard that the majority of the elite athletes couldn't satisfy.

    The mere fact that there is ONE woman in NASCAR already indicates that there is no direct biological barrier in possessing the necessary skills, regardless of whatever other reasons may be present.  Similarly with ONE woman in the NHL, this clearly indicates that a woman can possess the necessary skills to participate.  While you can argue about the numbers and the distributions, the fact is that even one argues against any claim that there is a biological barrier [beyond size and strength].
    Mundus vult decipi
    Gender baiting (anonymously) are we?

    I agree completely with your assertion that the ones who readily succeed in any classroom are indeed the very same ones favored by the teachers. I would not, however, pigeon-hole this very important systemic problematic abuse from a gender biased point of view. In addition to being gender biased it is also an issue of personality preferences and associated learning styles coupling up with appropriately compatible teaching styles.

    This article and study demonstrates the lack of the required unbiased objectivity in what has degenerated into what qualifies as "modern science."

    Any study suggesting that boys and girls are different is a completely wasteful exercise in the redundancy of the incredibly obvious. Any such study will be subject to the cancerous and notoriously vicious political correctness filter rather than be understood for what it is; incredibly obvious by simple inspection, but I suppose this is why this article was published anonymously.

    Only in a completely ridiculously idealistically biased philosophical perspective , I'll let which "ism" i'm speaking of here, could the "equality" of boys and girls ever be dreamed of let alone actually be asserted to as factual, let alone one that would need government money for a study to debunk the incredibly obvious.

    In short the need for the funding for scientific studies to point out obvious logical flaws in unscientific IDEALISMS is completely redundant and wasteful. However, I'm well aware of how well many "scientists" abuse "science" for their own personal gain.

    There is another area in "modern science" where this subjective (read NOT objective) practice runs rampant: Can you say "global warming?"

    First of all:

    If this data would indicate any biological difference between boys and girls, than it would be that girls are genetically better equipped for reading skills but have no disadvantage in mathematics (The reading gap in favor of girls is three times larger than the mathematics gap).

    Second of all:

    This data actually suggest, that there is no significant biological influence on reading and/or mathematic skills. Because as you can see in the kindly provided graphs girls started with time to overcome boys in some countries in mathematical skills which implicates a strong connection between educational and societal models and the success of girls in mathematical studies.

    Totally ridiculous from a scientific point of view is the reasoning in the first paragraph: Because we see a difference between girls and boys among almost any group of society, this difference has to be biological. As if gender is not an impact-factor in all groups of society. This is like saying in 1900: Because we see an underperformance of african americans in all parts of society, the reason for this has to be biological. It eludes me how people with a professorship can support such idiotic reasoning and still have a job.

    And those men and women who fail in life's pursuits survive by redoing the same study every ten years.

    I remember OTHER studies that said exactly the same thing. I don't think this is anything new.

    Gerhard Adam
    I'm surprised that no one has pointed out that humans [neither male or female] have not evolved to read or do math.  Leave an individual alone without education, and the genes won't ever exhibit a trait for reading or mathematics.

    However, something even more significant [and probably related] would be to explain the gender differences in guitarists.  One can hardly argue that there is an institutional bias. 
    Mundus vult decipi
    "Leave an individual alone without education, and the genes won't ever exhibit a trait for reading or mathematics."

    I disagree, while you are probably correct when referring to the conscious application of mathematics we all perform complex mathematics intuitively on a subconscious level. Anyone who watches their favorite wide receiver zero-in on and successfully catch a long pass can not deny that there has to be some impressive adaptive nonlinear calculus being easily performed by the old noggin.

    Almost anyone who has ever worked in artificial intelligence or robotics understands that our best technological efforts and even the entirety of our collective knowledge thus far are incredibly limited and crude.

    In addition to mathematics we are all also very capable of reading without any education. The alphabet is nothing more than a collection of contrived symbols who's usage is learned through repetitive education. Monkey see, monkey do. We all have a collective set of inherent archetypical symbols that we use to "read" each other, i.e. mother and new born baby have very little trouble "reading" each others symbolic gestures without being educated.

    Gerhard Adam
    Anyone who watches their favorite wide receiver zero-in on and successfully catch a long pass can not deny that there has to be some impressive adaptive nonlinear calculus being easily performed by the old noggin.
    Sorry, but such an assessment makes the entire concept of mathematics unintelligible.  After all, if that's the case, then to what do you attribute the ability of animals to preform similar actions with a great deal more accuracy?

    In actuality, you have it backwards.  The brain isn't performing "nonlinear calculus", but rather "nonlinear calculus" is what we use to describe what is occurring.

    Are we to assume that they've also studied physics?
    Mundus vult decipi
    "to what do you attribute the ability of animals to perform similar actions with a great deal more accuracy?"

    They also can perform very complex tasks that would require a machine with much greater technology than we currently possess in order to duplicate, and yes their brains also easily perform very complex mathematical tasks.

    The field of mathematics, as you most likely are referring to, is little more than a contrived construct using symbolisms to convey complex relationships. Very similar indeed to basic language but with a more narrowly defined purpose. Pure mathematics is the dialog and rhetoric of pure rational reason. Basic language, on the other hand, also includes and is often driven by the irrational. The construct, which is described and cataloged by this field, existed long before any human dreamed up its textual mathematical descriptions or symbols.

    That is why it has been said, quite correctly, that gravity did not exist until Newton described its effects. (Newton was very careful not to try to describe gravity, only its effects) The construct that allowed Newton to create his descriptions, however, existed long before Newton got beaned by the apple good, yeah, yeah ,yeah ,yeah.

    That is also why the arguments of idealists, who vainly attempt to include science, will nearly always end up with fatal flaws in their basic reasoning. Pointing them out is like shooting floating dead fish in a barrel, there is no sport in it, just a lot of satisfaction watching the cockroaches scurry when confronted with the light of a rational argument. This is also why an idealist will always make attempts at discrediting their opponent, this is their only line of defense since the content of their arguments rarely stand the tests of reason.

    Even an inanimate object follows with and is confined by the construct, NOT by our meager and incomplete descriptions of it. But any dog who can run and catch a Frisbee in mid air is NOT performing from simple muscle memory. In real time it must continuously calculate updates for convergent solutions and apply continuous corrections to its own trajectory in order to achieve its final slobbery solution.

    Life is pretty amazing, is it not? Much, much more so than any human technology. It is only in our own minds where we become gods and actually have any control over our own universe, physically or metaphysically.

    Gerhard Adam
    ...yes their brains also easily perform very complex mathematical tasks.

    But any dog who can run and catch a Frisbee in mid air is NOT performing from simple muscle memory. In real time it must continuously calculate updates for convergent solutions and apply continuous corrections to its own trajectory in order to achieve its final slobbery solution.
    I would appear that this is where the confusion begins.  A "mathematical task" is not mathematics.  That is merely what an external observer may use to describe what has occurred.  The individual brain performs no such calculations.

    By your description you're suggesting that the brain performs these calculations and yet possesses no conscious awareness that it is occurring.  More importantly, it bears no relationship to the ability to actually calculate using our symbolic methods of mathematics. 

    In fact, you would discover that the more engaged your brain is in the activity at hand [i.e. the more "calculating" is required] the greater the likelihood that the individual will fail in their activity [i.e. miss the ball].  This is precisely why an individual needs to practice as much as they do, to eliminate the "calculating" part of the brain's functions from participating, lest it occur too slowly and imprecisely.

    Mundus vult decipi
    You are correct that the conscious mind can and often does interfere with and suppresses the unconscious mind. I believe it is called a neurosis, and when the neurosis actually interferes with a persons normal activities it is called a psychosis.

    "A "mathematical task" is not mathematics. That is merely what an external observer may use to describe what has occurred. The individual brain performs no such calculations."

    Not true, if it were please point me to the practical machine you are speaking of that possesses only a conscious will (no mathematically intensive mechanisms or micro-controllers) to run and successfully catch a Frisbee. For that matter show me one that can perform this task equally given any current technology. Even a washing machine wouldn't function if it could not perform simple calculations, such as a simple comparator for testing when it should perform its next task.

    Performing mathematics is not, nor has ever been, solely limited to human activity.

    Gerhard Adam
    The point remains that running and successfully catching a Frisbee or hitting a baseball, etc. all involve a significant amount of practice.  This is precisely why we don't see such skills as innate.

    There may be some rudimentary capabilities acquired as one gets older and is capable of exercising finer muscle control, but it is not an innate capability.  Hand-eye coordination must be acquired.

    One can see this in all manner of activities where it is the continuous practice that trains the muscles and then allows an individual to respond in the fashion you're describing as "mathematical". 
    Even a washing machine wouldn't function if it could not perform simple calculations, such as a simple comparator for testing when it should perform its next task.
    Sorry, but that simply isn't true.  A thermostat operates without any calculating capability and yet it can respond to changes in temperature.  Of course, it's even more obvious when one examines the gears, and springs that govern the behavior of older clocks.  There are no "comparator" operations required, the mechanics of the system are sufficient without requiring any calculation on the part of the system.  Even your washing machine doesn't require a comparator operation, nor does it require a calculation. 

    You're stretching the definition of calculation and mathematics well beyond any reasonable meaning.
    Mundus vult decipi
    "A thermostat operates without any calculating capability and yet it can respond to changes in temperature."

    Sorry, simply not true. The thermostat does not perform the calculation, the HVAC controller does, based on the data from the thermostat. On a simple mechanical system the controller waits for the signal from the mercury switch, located in the thermostat, to change its logical state. The controller performs a simple mathematical logical calculation; the thermostat signal is either above or below a certain predetermined level, it is either a logical 1 or a logical 0. The circuit is designed with the deliberate mathematical logic decision making capabilities firmly in place, these circuits are generally called comparators.

    As far as your example with an old watch, the comparator function is still performed. This time by a spring pendulum and a ratcheting cog system of gears and levers. The simple clock also makes it a very valuable integrating device displaying the cumulative passage of time. That is correct a bucket of sand with a hole in it can perform calculus with ease.

    Last time I checked these are indeed inanimate objects performing mathematical functions.